string length scale (changes) in nLab

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Contents

Idea

In perturbative string theory, the string length scale ℓ s\ell_s is a unit of length which sets the scale for the extension of strings described by sigma-model worldsheet field theories. Specifically, the Nambu-Goto action for the string is nothing but the relativistic volume-functional, and the string length determines in which units the volume is measured.

The square of the string length is known as the Regge slope and (as traditionally in: denoted byα ′\alpha^\primeslope of Regge trajectories) and traditionally denoted by α ′\alpha^\prime:

α ′=ℓ s 2. \alpha^\prime \;=\; \ell_s^2 \,.

The inverse of the string length squared/Regge slope is called the string tension, traditionally denoted

T s=12πα ′=12πℓ s 2. T_s \;=\; \frac{1}{2\pi \alpha^\prime} = \frac{1}{2 \pi \ell_s^2} \,.

This way the Nambu-Goto action for the string with proper units attached is

L NG=Tvol Σ, L_{NG} \;=\; T vol_{\Sigma} \,,

where vol Σvol_{\Sigma} is the (induced) volume form on the worldsheet Σ\Sigma.

Properties

Relation to Planck length and string coupling

For discussion of relation to Planck length and string coupling constant see at non-perturbative effect the section Worldsheet and brane instantons

Vanishing tension limit

In the limit T s→0T_s \to 0, ℓ s→∞\ell_s \to \infty of vanishing string tension, string field theory is supposed to become Vasiliev’s higher spin gauge theory. See there for more.

• Regge slope

fundamental scales (fundamental/natural physical units)

• speed of light \, cc

• Planck's constant \, ℏ\hbar

• gravitational constant \, G N=κ 2/8πG_N = \kappa^2/8\pi

• Planck scale

  • Planck length \, ℓ p=ℏG/c 3\ell_p = \sqrt{ \hbar G / c^3 }

  • Planck mass \, m p=ℏc/Gm_p = \sqrt{\hbar c / G}

• depending on a given mass mm

  • Compton wavelength \, λ m=ℏ/mc\lambda_m = \hbar / m c

  • Schwarzschild radius \, 2mG/c 22 m G / c^2

  • depending also on a given charge ee

    • Schwinger limit \, E crit=m 2c 3/eℏE_{crit} = m^2 c^3 / e \hbar

• GUT scale

• string scale

  • string tension \, T=1/(2πα ′)T = 1/(2\pi \alpha^\prime)

  • string length scale \, ℓ s=α′\ell_s = \sqrt{\alpha'}

  • string coupling constant \, g s=e λg_s = e^\lambda

References

For instance:

• David Tong, around p. 16 in Lectures on String Theory (arXiv:0908.0333)